Keegan L. A. Kirk

Harbir Antil, Keegan L. A. Kirk, Felipe Pérez

We introduce and analyze Darcy flow through a saturated porous medium subject to bilateral constraints on the normal flux across a portion of the boundary. The problem is posed as the maximization of a velocity-based dual concave energy over a convex subset of $H(\mathrm{div};Ω)$; Fenchel duality identifies a pressure-based predual formulation, yields strong duality, and provides convex optimality conditions with a complementarity structure on the constrained boundary. The primal--dual gap satisfies an a posteriori error identity, free of generic constants, valid for arbitrary admissible approximations. The duality structure is inherited by a Raviart--Thomas/Crouzeix--Raviart discretization, from which we derive a discrete error identity and a priori error decay rates under fractional regularity assumptions on the solution and the flux bounds. Numerical experiments, including adaptive refinement driven by localized primal--dual gap indicators, support the theory.

Adrian Celaya, Keegan Kirk, David Fuentes et al.

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods like PINNs rely on auto differentiation and sampling collocation points, leading to a lack of interpretability and lower accuracy than traditional numerical methods. As a result, we propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.

Harbir Antil, Alex Kaltenbach, Keegan L. A. Kirk

A finite element discretization of an optimal insulation problem with convective heat transfer is considered. The model is formulated as a non-smooth, two-variable convex minimization problem. It accounts for the temperature distribution in a thermally conducting body $Ω\subseteq\mathbb{R}^d$, with $d\in \{2,3\}$, and the distribution of a given amount of insulation material on an insulated boundary part $Γ_I\subseteq \partialΩ$. The surface integral over the insulated boundary $Γ_I$ is approximated by a mass-lumping quadrature that preserves the structure of the continuous setting and, in particular, yields discrete optimality conditions mirroring their continuous counterparts. Well-posedness, stability, and weak convergence of discrete solutions to the continuous ones are established. Furthermore, a block coordinate descent algorithm for the computation of the discrete solutions is formulated and its linear convergence is derived. Under suitable regularity assumptions, uniform $L^\infty(Γ_I)$-bounds and $\textit{a priori}$ error estimates for both the temperature distribution and the distribution of a given amount of insulation material are obtained. Numerical experiments are carried out that confirm the predicted error decay rates and demonstrate the method in a qualitative three-dimensional test on a realistic spacecraft crew module capsule geometry with idealized reentry-heating Robin data.